Well-posedness and porosity in best approximation problems
type: Item
instance of: scholarly article
title: Well-posedness and porosity in best approximation problems
zbMATH Open document ID: 1005.41011
DOI: 10.12775/TMNA.2001.041
author: Simeon Reich
author: Alexander J. Zaslavski
published in: Topological Methods in Nonlinear Analysis
publication date: 2003-02-05T00:00:00Z
review text: In general a closed subset \(A\) of a Banach space \(X\) does not contain a best approximation to a point \(x\in X-A\). However, there are Banach spaces in which the set of such points,\(x\), which have best approximations in \(A\), form a dense subset (or more) of \(X\). The development of the field includes proving equivalent properties of the Banach spaces, determining the size of the set of points that have best approximations, and considering uniqueness of best approximations and the convergence of minimizing sequences. This work advances the theory by changing the framework to allow the set \(A\) to vary. The results apply to an arbitrary Banach space \(X\). There are three types of results. We describe simple forms of two of them. The first fixes the point \(x\), and defines a complete metric on the set \(S\) of closed nonempty subsets of \(X\). The second, extends this metric to \(S\times X\). The authors show that, in both metric spaces, the points that represent a setting with a unique best approximation form a dense set (in fact a set with a \(\sigma\)-porous complement).
reviewed by: Daniel Wulbert
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A52
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 1799630
zbMATH Keywords: best approximation
zbMATH Keywords: closed set
zbMATH Keywords: Banach space
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: first category
MaRDI profile type: Publication
full work available at URL: tmna.2001.041
OpenAlex ID: W2548117697
description: scientific article; zbMATH DE number 1799630